horizontal categorification



Horizontal categorification or Oidification describes the process by which

  1. a concept is realized to be equivalent to a certain type of category with a single object;

  2. and then this concept is generalized – or oidified – by passing to instances of such types of categories with more than one object.


  • This is to be contrasted with vertical categorification.

  • It can be argued that the term ‘categorification’ should be reserved for vertical categorification, since we can use ‘oidification’ for the horizontal concept.

  • It has rightly been remarked that groupoids are more fundamental than groups, algebroids are more fundamental than algebras, etc. Hence in a better world, the suffix would be characterizing the one-object special cases, not the general concepts.


David Roberts: How do Lie algebroids fit into this framework?

Urs Schreiber: at a rough level it is clear that the base space of a Lie algebroid has to be regardedas the “space of objects”. Certainly a Lie algebroid over a point is precisely a Lie algebra.

But for a more precise statement one needs a more conceptual way to think of Lie algebroids. I am claiming at ∞-Lie algebroid that there is a way to regard Lie algebroids precisely as certain kinds of synthetically smooth groupoids, namely those all whose morphisms have “infinitesimal extension” in some sense. In such a picture Lie algebroids are on the same footing as Lie groupoids and are precisely the many-object version of Lie algebras = infinitesimal Lie groups.

Mike: Is there (and do we want there to be) a general rule about whether an X-oid means an internal category whose one-object version is an X, or an enriched category whose one-object version is an X? The examples above seem to be taking the “internal” side, but the Cafe discussion on “ringoids” was about the “enriched” version; the two are very different! And “dg-algebroid” was suggested for dg-category, which is an enriched oidification of a dg-algebra, but a Hopf algebroid is an internal oidification of a Hopf algebra.

Urs: good point. We should say this at the beginning and split the list of examples in two sorts

Further discussion

Related nn-Café-discussion is in

Last revised on March 6, 2012 at 09:11:06. See the history of this page for a list of all contributions to it.